Games With Incomplete Information: Bayesian Nash …hrtdmrt2/Teaching/GT_2017_19/L8.pdfGames With...

Games With Incomplete Information: BayesianNash Equilibrium

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

June 29th, 2016

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Private vs. Public Information

2 Bayesian game

3 How do we model Bayesian games?

4 Bayesian Nash equilibrium

5 Exercises


Private vs. Public Information

On the Agenda


2 Bayesian game



5 Exercises



Introduction

I In many game theoretic situations, one agent is unsure about the payoffs orpreferences of others

I Examples:

- Auctions: How much should you bid for an object that you want, knowing thatothers will also compete against you?

- Market competition: Firms generally do not know the exact cost of theircompetitors

- Signaling games: How should you infer the information of others from the signalsthey send

- Social learning: How can you leverage the decisions of others in order to makebetter decisions

C. Hurtado (UIUC - Economics) Game Theory 1 / 17


Introduction

I We would like to understand what is a game of incomplete information, a.k.a.Bayesian games.

I First, we would like to differentiate private vs. public information.

I Example: Batle of Sex (BoS): "Coordination Game" (public information)

In Sequential BoS, all information is public, meaning everyone can see all the sameinformation:




I In this extensive-form representation of regular BoS, Player 2 cannot observe theaction chosen by Player 1.

I The previous is a game of imperfect information because players are unaware ofthe actions chosen by other player.

I However, they know who the other players are hat their possible strategies/actionsare. (The information is complete or public)

I Imagine that player 1 does not know whether player 2 wishes to meet or wishes toavoid player 1. Therefore, this is a situation of incomplete information, alsosometimes called asymmetric or private information.


Bayesian game

On the Agenda


2 Bayesian game



5 Exercises


Bayesian game

Bayesian game

I In games of incomplete information players may or may not know some informationabout the other players, e.g. their "type", their strategies, payoffs or preferences.

I Example: Tinder BoSPlayer 1 is unsure whether Player 2 wants to go out with her or avoid her, andthinks that these two possibilities are equally likely. Player 2 knows Player 1’spreferences. So Player 1 thinks that with probability 1/2 she is playing the gameon the left and with probability 1/2 she is playing the game on the right.


Bayesian game

Bayesian game

I In games of incomplete information players may or may not know some informationabout the other players, e.g. their "type", their strategies, payoffs or preferences.

I Example: Tinder BoSPlayer 1 is unsure whether Player 2 wants to go out with her or avoid her, andthinks that these two possibilities are equally likely. Player 2 knows Player 1’spreferences. So Player 1 thinks that with probability 1/2 she is playing the gameon the left and with probability 1/2 she is playing the game on the right.

I This is an example of a game in which one player does not know the payoffs of theother.


Bayesian game

Bayesian game

I More examples:

- Bargaining over a surplus and you aren’t sure of the size

- Buying a car of unsure quality

- Job market: candidate is of unsure quality

- Juries: unsure whether defendant is guilty

- Auctions: sellers, buyers unsure of other buyers’ valuations

I When some players do not know the payoffs of the others, a game is said to haveincomplete information. It’s also known as a Bayesian game.


Bayesian game

Bayesian game

I Example: First-price auction (game with incomplete information)

1. I have a copy of the Mona Lisa that I want to sell for cash

2. Each of you has a private valuation for the painting, only known to you

3. I will auction it off to the highest bidder

4. Everyone submits a bid (sealed → simultaneous)

5. Highest bidder wins the painting, pays their bid

6. If tie, I will flip a coin


Bayesian game

Bayesian game

I Example: Second-price auction (game with incomplete information)

1. I have a copy of the Mona Lisa that I want to sell for cash

2. Each of you has a private valuation for the painting, only known to you

3. I will auction it off to the highest bidder

4. Everyone submits a bid (sealed → simultaneous)

5. Highest bidder wins the painting, pays the second-highest bid

6. If tie, I will flip a coin


How do we model Bayesian games?

On the Agenda


2 Bayesian game



5 Exercises




I Formally, we can define Bayesian games, or "incomplete information games" asfollows:

- A set of players: I = 1, 2, · · · , n

- A set of States: Ω e.g. good or bad car.

- A signaling function that goes into type space and is one-to-one: τi : Ω→ Ti .

- Pure strategies that are profile of actions conditional on player’s type: σi : Ti → Ai

- Individual Utility of outcome y , given actions (a1, a2, · · · , an):

Ui (σ1, · · · , σn|ti ) =∑

a1

· · ·∑

an

ui (y , σ1(t1(ω)), · · · , σn(t1(ω))) · pi (y |ti (ω))

I What would be the BR of player i? maxσi Ui (σ1, · · · , σn|ti ) (not solvable)

I Player i needs to know what −i knows about him. Also, Player i needs to knowwhat i knows about −i . Moreover, i needs to know what −i know about himconditional on what i know about −i , and so on




I Harsanyi (1968) doctrine: There is a prior about the states fo the nature that iscommon knowledge

I This is also known as the common prior assumptionI Whit this assumption we can turn Bayesian games into games with imperfect

information.I This is a very strong assumption, but very convenient because any private

information is included in the description of the types.I With the common prior players can form beliefs about others’ type and each player

understands others’ beliefs about his or her own type, and so onI When players are not sure about the game they are playing you may consider:- Random events are considered an act of nature (that determine game structure)- Treat nature as another (non-strategic) player- Draw nature’s decision nodes in extensive formI Treat game as extensive form game with imperfect info: players may/may not

observe nature’s action




I Recall: BoS variant

Player 1 is unsure whether Player 2 wants to go out with her or avoid her, andthinks that these two possibilities are equally likely. Player 2 knows Player 1’spreferences. So Player 1 thinks that with probability 1/2 she is playing the gameon the left and with probability 1/2 she is playing the game on the right.

I Let’s put this into extensive form.




I BoS variant in extensive form:




I When players are not sure about other players’ preferences:

- Consider a game where each players has private information about his preferences.

- That can be model as ui (σi , σ−i , θi ) where θi ∈ Ti .

- Here we are assuming that θi is the type of player i .

- Note that we are assuming that each player knows its own type, but thatinformation is not public




I When players are not sure about other players’ preferences:

- An example of a game where players don’t know the preferences of the others canbe the one represented by the following normal form:

1\2 L RT 2θ1, 3θ2 1,1B 1,0 0,0

- Each player i knows his own type, but types are not public information


Bayesian Nash equilibrium

On the Agenda


2 Bayesian game



5 Exercises




I Bayesian Nash equilibrium is a straightforward extension of NE:

I Each type of player chooses a strategy that maximizes expected utility given theactions of all types of other players and that player’s beliefs about others’ types.

- Example: Let us consider the previous game:

1\2 L RT 2θ1, 3θ2 1,1B 1,0 0,0

- It is common knowledge among the two players that each player i’s type θi isindependently drawn from the uniform distribution on [0, 1].

- Let us derive a pure strategy Bayesian Nash Equilibrium in this game.




1\2 L RT 2θ1, 3θ2 1,1B 1,0 0,0

- We first note that player 1 has a dominant strategy to choose T when his type isθ1 >

12

- Player 2 has a dominant strategy to choose R when his type is θ2 <13 .

- We therefore conjecture the following form of equilibrium strategies

P1 :

T if θ1 ≥ θ∗1

B if θ1 < θ∗1

P2 :

L if θ2 ≥ θ∗2

B if θ2 < θ∗2

- Solving for the equilibrium requires solving for the constants θ∗1 and θ∗

2




1\2 L RT 2θ1, 3θ2 1,1B 1,0 0,0

- In a Nash Equilibrium each player must be indiferent between each of his purestrategies (Why?)

- player 1 plays T with probability 1− θ∗1 (Why?)

- player 2 plays L with probability 1− θ∗2 (Why?)

- Hence,2θ∗

1 · (1− θ∗2 ) + 1 · θ∗

2 = 1 · (1− θ∗2 ) + 0 · θ∗

2

3θ∗2 · (1− θ∗

1 ) + 0 · θ∗1 = 1 · (1− θ∗

1 ) + 1 · θ∗1

- From where we can determine that

θ∗1 = 1

6

θ∗2 = 2

5C. Hurtado (UIUC - Economics) Game Theory 16 / 17

Exercises

On the Agenda


2 Bayesian game



5 Exercises


Exercises

ExercisesI You and a friend are playing a 2× 2 matrix game, but you’re not sure if it’s BoS or

PD. Both are equally likely.

Put this game into Bayesian normal form.I Consider the following two person game of incomplete information:

1\2 L RT θ1, θ2 1, 1

2B 1

2 , 0 − 14 ,−

14

It is common knowledge among the two players that player 1’s type θ1 and player2’s type θ2 are independently drawn from the uniform distribution on [0, 1].Derive a pure strategy Bayesian-Nash equilibrium in this game.


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